Problem Analysis:
The problem involves vector addition to determine the resultant velocity of a plane relative to the ground. The given information is as follows:
1.
Plane Velocity (relative to the air):
- Magnitude: 100 km/hr
- Direction: Straight downward (as indicated by the arrow in the diagram).
2.
Wind Velocity:
- Magnitude: 25 km/hr
- Direction: Straight downward (as indicated by the arrow in the diagram).
3.
Resultant Velocity (relative to the ground):
- Magnitude: 125 km/hr
- Direction: Straight downward (as indicated by the arrow in the diagram).
The task is to verify or solve for the resultant velocity using vector addition principles.
---
Step-by-Step Solution:
#### 1. Represent the Velocities as Vectors:
-
Plane Velocity (\(\vec{V}_{\text{plane}}\)):
- Magnitude: 100 km/hr
- Direction: Downward (negative y-direction).
- Vector representation: \(\vec{V}_{\text{plane}} = -100 \hat{j}\) km/hr.
-
Wind Velocity (\(\vec{V}_{\text{wind}}\)):
- Magnitude: 25 km/hr
- Direction: Downward (negative y-direction).
- Vector representation: \(\vec{V}_{\text{wind}} = -25 \hat{j}\) km/hr.
#### 2. Add the Vectors to Find the Resultant Velocity:
The resultant velocity (\(\vec{V}_{\text{resultant}}\)) is the vector sum of the plane's velocity and the wind velocity:
\[
\vec{V}_{\text{resultant}} = \vec{V}_{\text{plane}} + \vec{V}_{\text{wind}}
\]
Substitute the vector representations:
\[
\vec{V}_{\text{resultant}} = (-100 \hat{j}) + (-25 \hat{j})
\]
Combine the components:
\[
\vec{V}_{\text{resultant}} = -125 \hat{j} \text{ km/hr}
\]
#### 3. Interpret the Result:
- The resultant velocity has a magnitude of 125 km/hr.
- The direction is straight downward (negative y-direction), consistent with the given information.
#### 4. Verify the Magnitude:
The magnitude of the resultant velocity is:
\[
|\vec{V}_{\text{resultant}}| = 125 \text{ km/hr}
\]
This matches the given magnitude of the resultant velocity.
---
Final Answer:
The solution confirms that the resultant velocity of the plane relative to the ground is indeed:
\[
\boxed{125 \text{ km/hr}}
\]
Parent Tip: Review the logic above to help your child master the concept of relative motion worksheet.
